Random Variable Random Variable – Numerical Result Determined by the Outcome of a Probability Experiment. Ex1: Roll a Die X = # of Spots X | 1 2 3 4 5 6 P(X) | 1/6 1/6 1/6 1/6 1/6 1/6 Ex2: Toss a Coin Twice X = # of Heads X | 0 1 2 P(X) | ¼ ½ ¼

Ex3: Inspect an Item X = Diameter Ex4: Select and Employee X = Tenure with Company Probability = The Area of the Random Variable Distribution Curve

Discrete Random Variable – Takes on Only Integer Values Continuous Random Variable – Takes on All Real Values Over a Range Measures for Random Variables Mean µ = E(X) = ∑ X•P(X) Expected Value 2 Coin Toss: X | 0 1 2 P(X) | ¼ ½ ¼

3 Coin Toss: X | 0 1 2 3 P(X) | 1/8 3/8 3/8 1/8 Variance of a Random Variable σ2 = V(X) = ∑ (X - µ )2•P(X) Variance σ2 = E(X2) - µ2 Shortcut Formula

2 Coin Toss: X | 0 1 2 P(X) | ¼ ½ ¼ (X-µ) | (X-µ)2 | (X-µ)2•P(X) | 3 Coin Toss: X | 0 1 2 3 P(X) | 1/8 3/8 3/8 1/8 (X-µ) | (X-µ)2 | (X-µ)2•P(X) |

Example: Insurance Claim X = $ Loss $ Loss P(X) X•P(X) (X-µ) (X-µ )2 (X-µ )2•P(X) 0 .938 2,000 .05 10,000 .01 50,000 .002

St Petersburg Paradox: Gambling Game in which you toss a coin until you get a Tail. # Tosses | 1 2 3 4 5 6 7 8 9 Payoff | $2 $4 $8 $16 $32 $64 $128 $256 $512 P(X) | ½ ¼ 1/8 1/16 1/32 1/64 1/128 1/256 1/512 Expected Payoff = Tosses =